32n 1 Is Divisible By 8 For All Natural Numbers

32n – 1 is divisible by 8, for all natural numbers .

Answers:

Let P(n): 3  – 1 is divisible by 8, for all natural numbers n.
Now, P(l): 3  – 1 = 8, which is divisible by 8.
Hence, P(l) is true.
Let us assume that, P(n) is true for some natural number n = k.
P(k): 32k – 1 is divisible by 8
or 32k -1 = 8m, m ∈ N  (i)
Now, we have to prove that P(k + 1) is true.
P(k+ 1): 32(k+1)– l
= 32k • 32 — 1
= 9(8m + 1) – 1     (using (i))
= 72m + 9 – 1
= 72m + 8
= 8(9m +1), which is divisible by 8 Thus P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for all natural numbers n.